A radian is defined as the angle of an arc the length of the radius of the circle. In your case the arc length is $^{350} / _{800} =\; ^7 / _{16}$ of the radius. From there the amount of radians should follow.
–
tescMar 4 '13 at 9:22

1 Answer
1

Perhaps you can draw a labelled picture. Let $O$ be the centre of the circle, let $P$ and $Q$ be the two points of tangency, and let $M$ be the point where the two tangents meet. It will turn out that the angle $POQ$ is not very far from $30$ degrees, so a realistic picture should have $\angle POQ$ of about that size.

One useful thing to remember is that the radius $OP$ is perpendicular to the tangent line $PM$ at $P$, and the radius $OQ$ is perpendicular to the tangent line at $Q$.

So the quadrilateral $OPMQ$ has two right angles. We have been asked for the angle at $M$. Since the angles of a (convex) quadrilateral add up to $360$ degrees, the angles at $M$ and at $O$ have sum $180$ degrees, that is, $\pi$ radians.

So the angle at $M$ is $\pi$ minus the angle at $O$.

The angle at $O$ is, in radians, $\dfrac{350}{800}$. This has been mentioned in the comments. There are various ways of remembering it. One way is to recall that all the way around the circle is a distance $(2\pi)(800)$. It is also a full rotation, so it is $2\pi$ radians. Thus $1$ foot along the circumference is $\dfrac{1}{800}$ of a radian, and therefore $350$ feet along the circumference is $\dfrac{350}{800}$ radians.

Finally, the angle at $M$ is therefore $\pi-\dfrac{350}{800}$ radians.